Spivak's Calculus chapter 1 problem 12 v

Question

I am having trouble proving $$|x|-|y|≤|x-y|.$$ In the solutions it says $$|x|=|y-(y-x)|≤|y|+|y-x|, \quad \text{so} \quad |x|-|y|≤|x-y|.$$ Am I missing something here? How did he get $|x-y|$ on the right side?

Answer 1

$|x|=|-x|$ where x is any real number.

so $|x-y|=|y-x|$

Answer 2

Note that $|x|=|-x|$ for any $x\in\mathbb R$.

Hence, we have $$|y-x|=|-(y-x)|=|x-y|.$$

So, $|x|\le |y|+|y-x|$ and $|y-x|=|x-y|$ lead $$|x|-|y|\le|x-y|.$$

Answer 3

we have used in your proof that $|y-x|=|x-y|$